
with \(\Phi \)  and \(\Psi \)  being scalar perturbations, and \(h_{ij}\)  the transverse-traceless tensor perturbation to the second-order.
We neglect the first-order contribution to the tensor perturbation below. Furthermore, we neglect the vector perturbation and the anisotropic stress tensor, and thus \(\Phi = \Psi \) .
The energy density of the gravitational waves in the subhorizon region is given by [1]}, [2]}
\(\rho _{\rm GW} = \frac{M_{\rm P}^2}{16a^2}\langle \overline{\partial _k h_{ij} \partial ^k h^{ij}}\rangle \,,\) 
